Inter-species topological phases via a dynamical gauge field

TL;DR

The study introduces inter-species topological phases in a two-species lattice coupled by a dynamical gauge field, revealing extrinsic edge-confined states and intrinsic bulk-bound states arising from distinct inter-species topologies. It demonstrates how inter-species band inversion activates the DGF to produce nonreciprocal hopping and long-range correlations, yielding PT-symmetry–related spectral features and characteristic dynamical signatures. A concrete Floquet-based cold-atom realization is proposed, with analytical and numerical tools to characterize PT transitions, bulk-bound state formation, and multi-particle extensions. The results establish inter-species topology as a new organizing principle for topological matter in hybrid quantum systems and open avenues for exploring correlation-enabled topological phenomena beyond single-particle paradigms.

Abstract

We uncover a class of inter-species topological phases in a one-dimensional lattice, loaded with two species of non-identical particles interacting via a dynamical gauge field (DGF). Two types of topological states are found to emerge from different inter-species topology activated by the DGF. Specifically, edge confined states with co-localization of both species arise from an extrinsic inter-species topology, which can be decomposed into the single-particle topology for each species. On the other hand, bulk bound states with extended distribution emerge from an intrinsic inter-species topology that cannot be understood from single-particle ones. The two classes of inter-species topology are found to be independent of each other, characterized by different sets of inter-species topological invariants. Thus, their topological states can coexist in certain parameter regimes and compete with each other, leading to distinguished dynamical signatures. We further propose a feasible cold-atom realization of our model to demonstrate experimental accessibility of inter-species topological phases. Our work establishes inter-species topology as a new organizing principle of topological matter, revealing how correlations between distinct particle species can generate topological phenomena beyond single-particle paradigms.

Figures (24)

• Figure 1: Inter-species topological phases and our model.a Topological band structures of conventional and inter-species topological phases and their topological states. Orange (green) circles with plus (minus) sign indicate the symmetry indicators at high-symmetric momenta, and topological band inversion is marked by dashed lines with arrows. Band inversion of a single species gives rise to the conventional topological phases. Extrinsic ISTPs emerge from the correlation between two species with and without single-species band inversion. Intrinsic ISTPs are induced by inter-species band inversion, regardless whether single-species band inversion occurs. b A sketch of the model of Eq.\ref{['eq:DGF_H']}. $\bar{\sigma},\sigma\in\{\uparrow,\downarrow\}$ with $\bar{\sigma}\neq\sigma$ denote the two species interacting with each other via a DGF ($F_{\sigma,2j}$).
• Figure 2: edge confined and anti-confined states.a OBC spectrum of the Hamiltonian in Eq. \ref{['eq:DGF_H']}, marked by the edge-density imbalance of the pseudospin-down particle. Gray dots are the eigenenergies with PBCs taken only for the pseudospin-down particle. b1 and c2 distributions of pseudospin-up and -down particles, respectively, for OBC eigenstates in the enlarged insets in a marked by the same colors, while b2 and c2 show these for the associated states with eigenvalues ${\rm Im}E\approx-\gamma_\uparrow$. Their average distributions are shown in black in each panel. edge confined states are characterized by the co-localization at the edge for both particles. Parameters in a to c2 are $v_{\uparrow}=5$, $v_{\downarrow}=0.5$, $\gamma_{\uparrow}=0.5$, $u_{\uparrow}=2$, and $u_{\downarrow}=1$. d to f2 the same as a to c2, but with $u_{\uparrow}=-2$ and $u_{\downarrow}=-1$. Edge anti-confined states are characterized by the pseudospin-up localization and the drop of pseudospin-down density at the same edge. In c1-c2 and f1-f2, distributions of single eigenstates and their average correspond to different y-axis coordinates on the right and left sides of the figure, respectively. The chosen system size is $L=32$, with $L^2=1024$ the Hamiltonian dimension.
• Figure 3: Bulk bound states induced by inter-species band inversion and DGF.a and b The PBC spectra of the Hamiltonian in Eq. \ref{['eq:DGF_H']} with different parameters. The normalized two-particle correction of bulk bound states [Eq. \ref{['Twocorr']}] with the highest entanglement entropy (marked by arrows) is shown in the insets. Parameters are a$\theta_{\uparrow}=0.1\pi$ and $\theta_{\downarrow}=0.9\pi$, and b$\theta_{\uparrow}=0.05\pi$ and $\theta_{\downarrow}=0.15\pi$, with $\theta_{\sigma}={\rm arg}(u_{\sigma}+iv_{\sigma})$, $|u_{\sigma}+iv_{\sigma}|=\sqrt{2}$, and $\gamma_\uparrow=0.5$. c Magnitude of the DGF term $M$ defined in Eq. \ref{['eq:DGF_matrix']}. d A phase diagram spanned by $\theta_{\uparrow}$-$\theta_{\downarrow}$ based on $\{I_{00},I_{\pi\pi},I_{0\pi}\}$ defined in Eq. \ref{['ISbi']}. Nontrivial regions with at least one $I_{kk'}=-1$ are marked by different colors, and shaded regions represent the single-particle $\mathcal{PT}$-broken phase of pseudospin-up particle (see Supplementary Note 4 C), where the topological invariants are ill-defined. Trivial regions marked by $\{+++\}$ have vanishing DGF ($M\approx 0$) in c. Additionally, dark and light blue (red and pink) regions possess edge confined (anti-confined) states induced by single-particle topology of the pseudospin-up and -down particles, respectively. e The maximal entanglement entropy $S_m$ of all eigenstates. Red square and diamond mark the parameters of a and b with the same symbols. f The same as e, but with an extra disorder term $\lambda\sum_{\sigma}\sum_j\varepsilon_{\sigma,j}n_{\sigma,j}$ with $\lambda=0.2$ and $\varepsilon_{\sigma,j}$ randomly drawn from a uniform distribution $[-1,1]$. $L=32$ is set for all panels.
• Figure 4: The dynamical competition between two types of bound state.a The largest imaginary energies for all PBC eigenstates, corresponding to that of bulk bound states in the inter-species nontrivial regimes. Black lines indicate the same phase boundaries as in Fig. \ref{['fig2']}. White box marks the region with $\mathcal{PT}$-broken edge confined states as in Fig. \ref{['fig1']}a to c. b The mean location of pseudospin-up and -down particles during the evolution process for the initial state $|\psi(0)\rangle=a^\dagger_{\uparrow,15}a^\dagger_{\downarrow,18}|0\rangle$. c The normalized two-particle correction $\widetilde{\Gamma}_{j,j'}$ of the evolved state at the end time $\tau=150$. Parameters in b and c are $\theta_{\uparrow}=0.6\pi$ and $\theta_{\downarrow}=0.1\pi$, marked by the red square in a. d The same as b, but with $\theta_{\uparrow}=0.45\pi$ and $\theta_{\downarrow}=0.2\pi$, marked by the red circle in a. e Distribution of pseudospin-down particle at several different time $\tau$, and its average during the time interval $\tau\in[50,150]$, with the same parameters as in d. In all panels, $|u_{\sigma}+iv_{\sigma}|=\sqrt{2}$, $\gamma_{\uparrow}=0.5$, and $L=32$ are set.
• Figure S 1: A sketch and eigensolutions of the effective model $H_{\downarrow,{\rm eff}}$. (a) Lattice structure of the Hamiltonian $H_{\downarrow,{\rm eff}}$ in Eq. \ref{['meanDO']}. Here $\eta=\kappa^2=u^2_{\uparrow}/v^2_{\uparrow}$ and $t'=t(1-\eta)$. (b1) The PBC (gray dots) and OBC (colored) spectra of$H_{\downarrow,{\rm eff}}$. The OBC spectrum is marked by colors according to the edge-density imbalance. (b2) Distributions of OBC eigenstates, marked by the same colors as in (b1). Their average distribution are shown by the black line and dots. Parameters in (b1-b2) are $v_{\uparrow}=5$, $v_{\downarrow}=0.5$, $\gamma_{\uparrow}=t = 0.5$, $u_{\uparrow}=2$, and $u_{\downarrow}=1$, i.e., same as those for Figs. 2(a-c) of the main text. (c1-c2) The same as (b1-b2), but with $u_{\uparrow}=-2$ and $u_{\downarrow}=-1$, which are the same as those for Figs. 2(d-f) of the main text. The features presented in (b1-b2) resemble the edge confined states, while those in (c1-c2) resemble the edge anti-confined states.